284-286] 



DYNAMICAL EQUATIONS. 



515 



in like manner (dp zx /dz) $z . 8xSy. Hence, with our usual 

 notation, 



Y , up xx . up yx upzx 



)A. -\ j 1 -. 1 j , 



ax dy dz 



Pn* = 



_ 

 P ~ 



'dz' 



.(1). 



dy 



Substituting the values of p xxj p xy , ... from Art. 284 (4), (5), 

 we find 



Du dp d6 



p=pX - + ^ + ^ U ' 



Dv 

 'Di 



Dw 



dp d6 

 dj + ^dy 



dp M 



(2), 



where 



,, _ du dv dw 

 dx dy dz" 



and V 2 has its usual meaning. 



When the fluid is incompressible, these reduce to 



(3), 



.(4). 



These dynamical equations were first obtained by Navier* 

 and Poisson-|- on various considerations as to the mutual action of 

 the ultimate molecules of fluids. The method above adopted, 

 which is free from all hypothesis of this kind, appears to be due 

 in principle to de Saint-Venant and Stokes}. 



* "M6moire sur les Lois du Mouvement des Fluides," Mem. de I'Acad. des 

 Sciences, t. vi. (1822). 



t "M6moire sur les Equations g6n6rales de l'6quilibre et du Mouvement des 

 Corps solides elastiques et des Fluides," Journ. de VEcole Polytfchn., t. xiii. (1829). 



I " On the Theories of the Internal Friction of Fluids in Motion, &c.," Camb. 

 Trans., t. viii. (1845); Math, and Phys. Papers, t. i., p. 88. 



332 



